ADO inequality for combining three quantum states

Let $\mathcal{D}(d)$ denote the set of $d$-dimensional quantum states. Let $f:\mathcal{D}(d)\to\mathbb{R}$ be concave and symmetric, let $\rho_1,\rho_2,\rho_3\in\mathcal{D}(d)$ be states, and let $(p_1,p_2,p_3)$ be a probability distribution. Let $\rho$ be the combined state given by the constructions in equations (p:1), (p:2), and (p:3), with the parameters $p_i$ and $\delta_{ij}$ subject to equation (eq:deltas). ADO inequality.

$$f(\rho)\geq p_1f(\rho_1)+p_2f(\rho_2)+p_3f(\rho_3).$$

This conjecture proposes the three-state generalization of the entropy power inequality proved for two states, and would establish the corresponding lower bound for every concave symmetric function. The source identifies it as the main open problem, and no resolution is supplied here.